In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V (G) is a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality of a double dominating set of G is the double domination number γ×2(G). If G 6= C5 is a connected graph of order n with minimum degree at least 2, then we show that γ×2(G) ≤ 3n/4 and we characterize those graphs ac...

We characterize the edge-signed graphs in which every 'significant' positive closed walk (or combination of walks) has even length, under seven different criteria for significance, and also those edge-signed graphs whose double covering graph is bipartite. If the property of even length is generalized to positivity in a second edge signing, the characterizations generalize as well. We also char...

Diagrams and an Opfibration Account of Typed Graph Transformation R. Banach Computer Science Dept., Manchester University, Manchester, M13 9PL, U.K. [email protected] A. Corradini Dipartimento di Informatica, Università di Pisa, Corso Italia 40, Pisa, Italy. [email protected] Abstract: The “in the large” properties of typed graph transformation systems in the double pushout framework and a d...

A subset S of {0, 1, . . . , 2t − 1}n is called a t-fold MDS code if every line in each of n base directions contains exactly t elements of S. The adjacency graph of a t-fold MDS code is not connected if and only if the characteristic function of the code is the repetition-free sum of the characteristic functions of t-fold MDS codes of smaller lengths. In the case t = 2, the theory has the foll...

A vertex of a graph is said to dominate itself and all of its neighbors. A double outer-independent dominating set of a graph G is a set D of vertices of G such that every vertex of G is dominated by at least two vertices of D, and the set V (G) \D is independent. The double outer-independent domination number of a graph G, denoted by γ d (G), is the minimum cardinality of a double outer-indepe...

For a graph G with vertex set V(G) = {v1, v2, . . . , vn}, the extended double cover G∗ is a bipartite graph with bipartition (X, Y), X = {x1, x2, . . . , xn} and Y = {y1, y2, . . . , yn}, where two vertices xi and yj are adjacent if and only if i = j or vi adjacent to vj in G. The double graphD[G] of G is a graph obtained by taking two copies of G and joining each vertex in one copy with the n...

A complete classification is given of vertex primitive and vertex bi-primitive s-arc regular graphs with s ≥ 3. In particular, it is shown that the Petersen graph and Coxeter graph are the only vertex primitive 3-arc regular graphs, and that vertex bi-primitive 3-arc regular graphs consist of the complete bipartite graph K3,3, the standard double covers of the Petersen graph and Coxeter graph, ...

Graphical matrices are presented. Their construction via selected sets of subgraphs and the replacement of subgraphs by numbers representing graph invariants are discussed. The last step of the procedure is to apply the method of choice for obtaining the desired double invariant from the graphical matrix in the numerical form. It is also pointed out that many so-called special graph-theoretical...

The strong embedding conjecture states that every 2-connected graph has a closed 2-cell embedding in some surface , i . e . an embedding that each face is bounded by a circuit in the graph . A graph is called k -crosscap embeddable if it can be embedded in the surface of non-orientable genus k . We confirm the strong embedding conjecture for 5-crosscap embeddable graphs . As a corollary , every...

We prove that a cubic 2-connected graph which has a 2-factor containing exactly 4 odd cycles has a cycle double cover. © 2004 Elsevier Inc. All rights reserved. MSC: 05C38; 05C40; 05C70